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Differential operators of the second order and harmonic analysis. (Differentialoperatoren zweiter Ordnung und Harmonische Analysis.) (German) Zbl 0891.22008
In this very nice paper the author gives a review of methods from non-commutative harmonic analysis in the study of partial differential operators. For simplicity he restricts to second-order differential operators of the form $$L = - \sum_{i,j=1}^m a_{ij}X_i X_j +$$ lower order terms, where the $$X_j = \sum_{k=1}^d b_{jk}\partial / \partial x_k$$ are smooth vector fields on a $$d$$-dimensional smooth manifold $$M$$ for all $$j \in \{ 1,\ldots,m \}$$ and $$m \leq d$$. Moreover, the matrix $$(a_{ij})$$ is supposed to be real and symmetric. The operator $$L$$ is called transversally elliptic in case the matrix $$(a_{ij})$$ is positive, or negative, definite. The author discusses three main problems related to the operator $$L$$. The first is the regularity of the operator $$L$$. If the vector fields satisfy the Hörmander condition of order $$r$$, i.e., the vector fields $$X_1,\ldots,X_m$$, together with the commutators of order at most $$r$$ span the tangent space at every point of $$M$$, then every transversally elliptic operator $$L$$ is hypoelliptic according to a paper of L. Hörmander [Acta Math. 119, 147-171 (1967; Zbl 0156.10701)]. More detailed and quantitative regularity results have been obtained by L. P. Rothschild and E. M. Stein [Acta Math. 137, 247-320 (1976; Zbl 0346.35030)] via a lifting to a (in general non-commutative) nilpotent Lie group. This is illustrated by an example. The second problem concerns the local solvability of the operator $$L$$. This problem has not been solved in general, but for classes of differential operators in which $$L$$ occurs as a left invariant differential operator on a nilpotent Lie group the problem has been solved. It turns out that the local solvability also depends on the lower order terms. For second-order left invariant differential operators on nilpotent Lie groups with rank two the problem was solved by the author and F. Ricci [Math. Ann. 304, 517-547 (1996; Zbl 0848.43008)]. If $$L$$ is essentially self-adjoint on $$L_2(M)$$ and $$m \in L_\infty({\mathbb{R}})$$ then $$m(L)$$ is a bounded operator on $$L_2(M)$$. The third problem is to find conditions which ensure that the operator $$m(L)$$ maps $$L_2(M) \cap L_p(M)$$ into $$L_p(M)$$ and extends to a bounded operator on $$L_p(M)$$. In particular, does there exist a Marcinkiewicz-Mikhlin-Hörmander type theorem? For sublaplacians on stratified Lie groups such a theorem exists, but the best condition on the constants is still an open problem.
##### MSC:
 22E30 Analysis on real and complex Lie groups 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 43A80 Analysis on other specific Lie groups